Chapter 15: Decisions Under

Risk and Uncertainty

McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
In business we are forced to
make decisions involving
risk—that is, where the
consequences of any action
we take is uncertain due to
unforeseeable events. Where
do we begin?

15-2
 Risk vs. Uncertainty
• Risk
• Must make a decision for which the
outcome is not known with certainty
• Can list all possible outcomes & assign
probabilities to the outcomes
• Uncertainty
• Cannot list all possible outcomes
• Cannot assign probabilities to the outcomes

15-3
The game of roulette
is subject to risk. The
return on investment
in R & D is subject to
uncertainty.

15-4
 Analyzing the Problem—Preliminary Steps

• Listing the available alternatives, not only for direct

action but also for gathering information on which to
base later action;
• Listing the outcomes that can possibly occur (these
depend on chance events as well as on the decision
maker’s own actions);
• Evaluating the chances that any uncertain outcome will
occur; and
• Deciding how well the decision maker likes each
outcome.

15-5
 These are Experiments
processes that
generate well-
defined
outcomes
Experimental
Experiment Outcomes
Toss a coin Head, tail
Select a part for Defective, nondefective
inspection
Conduct a sales call Purchase, no purchase
Roll a die 1, 2, 3, 4, 5, 6
Play a football game Win, lose, tie

15-6
 Probability is a numerical
measure of the likelihood
of an event occurring

0 1.0
0.5
Probability:

The occurrence of the event is just

as likely as it is unlikely 15-7
 Unfortunately, business
experiments are rarely
repeatable. We cannot introduce
a new product 1,000 times and
measure the frequency with
which it succeeds—can we?

15-8
 Objective versus Subjective Probabilities

• Repeatable experiments (tossing a die,

flipping a coin) generate objective
probabilities.
• Non-repeatable experiments necessarily
involve assigning hypothetical or
subjective probabilities to particular
outcomes.

15-9
According to the subjective view, the probability of an
outcome represents the decision maker’s degree of
belief that the outcome will occur.

I estimate my odds of
becoming a country
music star at
two to one.

15-10
 Measuring Risk with
Probability Distributions
• Table or graph showing all possible
outcomes/payoffs for a decision & the
probability each outcome will occur
• To measure risk associated with a
decision
• Examine statistical characteristics of the
probability distribution of outcomes for the
decision

15-11
Probability Distribution for Sales
(Figure 15.1)

15-12
 Expected Value
• Expected value (or mean) of a
probability distribution is:
n
E( X )  Expected value of X   pi X i
i 1

Where Xi is the ith outcome of a decision,

pi is the probability of the ith outcome, and
n is the total number of possible outcomes

15-13
 Expected Value
• Does not give actual value of the
random outcome
• Indicates “average” value of the outcomes if
the risky decision were to be repeated a
large number of times

15-14
(Subjective) Probability Distribution for a New
Product Launch

The following gives the subjective view of a

manager concerning the probability distribution for
the first year’s outcome of a new product launch.

Outcome Sales Revenue Probability

Complete success $10,000,000 0.1
Promising 7,000,000 0.3

Mixed response 3,000,000 0.2

Failure 1,000,000 0.4

15-15
 Computing Expected Value or E(v)
Suppose the decision maker faces a risky prospect than
has n possible monetary outcomes, v1, v2, . . . , vn,
predicted to occur with probabilities p1, p2, . . . , pn. Then
the (monetary) expected value is found by:

E (v)  p1v1  p2v2  ...  pn vn

E(v) for the new product launch is given by:

E (v)  (0.1)($10)  (0.3)($7)  (.2)($3)  (0.4)($1)  $4.1 million

15-16
The Wildcatters Drilling Problem
Payoffs measured in
1,000s Wet
$600
0.4
Drill
120

Dry
120 -$200
0.6

Do not drill
$0

15-17
The Expected Value Criterion
I will select the course of action
with the highest expected value.
Note that the expected value of
“not drilling” is equal to zero.

Expected value of the drilling option:

E (v)  (0.4)(600,000)  (0.6)( 200,000)  $120,000

15-18
Good Decision, Bad Outcome

OK, the well turned out to be dry.

But that does not mean the
decision to drill was a bad one.
When confronted with risk, we
must distinguish between bad
decisions and bad outcomes.

15-19
 My expected value
of homes sold this
month is 1.246

Expected values for

discrete random
variables can assume
any value.

15-20
 Variance
• Variance is a measure of absolute risk
• Measures dispersion of the outcomes about
the mean or expected outcome
n
Variance(X) =  X2   pi ( X i  E( X ))2
i 1

• The higher the variance, the greater the

risk associated with a probability
distribution
15-21
Identical Means but Different
Variances (Figure 15.2)

15-22
 Standard Deviation
• Standard deviation is the square root of
the variance

 X  Variance( X )

• The higher the standard deviation, the

greater the risk

15-23
 Probability Distributions with
Different Variances (Figure 15.3)

15-24
 Coefficient of Variation
• When expected values of outcomes differ
substantially, managers should measure
riskiness of a decision relative to its
expected value using the coefficient of
variation
• A measure of relative risk

Standard deviation 
 
Expected value E( X )

15-25
 Decisions Under Risk
• No single decision rule guarantees
profits will actually be maximized
• Decision rules do not eliminate risk
• Provide a method to systematically include
risk in the decision making process

15-26
 Summary of Decision Rules
Under Conditions of Risk
Expected Choose decision with highest expected value
value rule
Mean- Given two risky decisions A & B:
variance • If A has higher expected outcome & lower
rules variance than B, choose decision A
• If A & B have identical variances (or standard
deviations), choose decision with higher
expected value
• If A & B have identical expected values, choose
decision with lower variance (standard deviation)
Coefficient of Choose decision with smallest coefficient of
variation rule variation

15-27
Probability Distributions for Weekly Profit
at Three Restaurant Locations
E(X) = 3,500 E(X) = 3,750
A = 1,025 B = 1,545
 = 0.29  = 0.41

E(X) = 3,500
C = 2,062
 = 0.59

15-28
 Which Rule is Best?
• For a repeated decision, with identical
probabilities each time
• Expected value rule is most reliable to
maximizing (expected) profit
• Average return of a given risky course of
action repeated many times approaches
the expected value of that action

15-29
 Which Rule is Best?
• For a one-time decision under risk
• No repetitions to “average out” a bad
outcome
• No best rule to follow
• Rules should be used to help analyze &
guide decision making process
• As much art as science

15-30
 Expected Utility Theory
• Actual decisions made depend on the
willingness to accept risk
• Expected utility theory allows for
different attitudes toward risk-taking in
decision making
• Managers are assumed to derive utility
from earning profits

15-31
 Risk Aversion
When it comes to risks that are
large relative to financial
resources, firms and individuals
tend to adopt a conservative
attitude. That is, they are not
“risk-neutral”—which is what the
expected value criterion assumes.

15-32
A Coin Gamble
You have 2 choices: You can have
$60, no questions asked. Or, you can
accept the following gamble: A fair
coin is tossed. Heads: you win $400.
Tails: You lose $200 dollars. OK, what
is your choice?

Question: Which choice has the highest

expected value?
For the “gamble” choice:
E (v)  (0.5)( 400)  (0.5)( 200)  $100

If you refuse the bet, you are not risk neutral!

15-33
 The Certainty Equivalent (CE)

The CE is the amount of

money for certain that
makes the individual
exactly indifferent to the
risky prospect.

Example: Suppose that a guaranteed $25 would make you

indifferent to the bet with an expected value of $100. Thus your
CE = $25. If CE < E(v), then the individual is risk averse.

15-34
The CE and the Degree of Risk Aversion
Principle: The higher the discount factor, the
higher the degree of risk aversion.

The discount factor for risk is the

difference between the expected value
of the gamble and the CE. My CE is
$75. Thus my discount factor is equal
to: $100 - $75 = $25. If your CE is
$25, then your discount factor is $75.

15-35
The Demand For Insurance
Insurance entails the transfer (for a price) of risk
from risk averse individuals or firms to risk neutral
insurance companies.

Risk pooling allows insurance

companies to be risk neutral. An
insurance company does not know
which homes will catch fire. It can
predict how many homes out of 10,000
will catch fire, however.

15-36
 Expected Utility
This is a similar decision making
process to expected value—with one
big difference. In contrast to the risk
neutral manager, who averages
monetary values at each step, the
risk averse manager averages
expected utilities associated with
monetary values.

15-37
 Expected Utility Theory
• Managers make risky decisions in a way
that maximizes expected utility of the
profit outcomes
E U(  )  p1U( 1 )  p2U( 2 )  ...  pnU( n )

• Utility function measures utility associated

with a particular level of profit
• Index to measure level of utility received for a
given amount of earned profit
15-38
Manager’s Attitude Toward Risk
• Risk averse
• If faced with two risky decisions with equal
expected profits, the less risky decision is
chosen
• Risk loving
• Expected profits are equal & the more risky
decision is chosen
• Risk neutral
• Indifferent between risky decisions that
have equal expected profit
15-39
Manager’s Attitude Toward Risk
• Determined by the manager’s marginal
utility of profit:
MU profit  U(  ) 

• Marginal utility (slope of utility curve)

determines attitude toward risk

15-40
Manager’s Attitude Toward Risk
• Can relate to marginal utility of profit
• Diminishing MUprofit
• Risk averse

• Increasing MUprofit
• Risk loving

• Constant MUprofit
• Risk neutral
15-41
 Expected Utility of Profits
• According to expected utility theory,
decisions are made to maximize the
manager’s expected utility of profits
• Such decisions reflect risk-taking attitude
• Generally differ from those reached by
decision rules that do not consider risk
• For a risk-neutral manager, decisions are
identical under maximization of expected utility
or maximization of expected profit

15-42
Manager’s Attitude Toward Risk
(Figure 15.5)

15-43
Manager’s Attitude Toward Risk
(Figure 15.5)

15-44
Manager’s Attitude Toward Risk
(Figure 15.5)

15-45
 A Risk Averse Wildcatter
Notes:
• U($600) = 100
• U (-$200) = 0

15-46
 Expected Utility: How It Works
•The decision maker first attaches a utility value to each
possible monetary outcome.
•The worst monetary outcome is assigned a value of zero;
the best monetary outcome must have a greater than
zero—but that is the only requirement.
•Our wildcatter assigns a value of 0 to the worst outcome
(-$200) and a value of 100 to the best outcome ($600).
Thus the expected utility of drilling is given by:

E (u )  (0.4)U (600)  (0.6)U (200)

 (0.4)(100)  (0.6)(0)  40.

15-47
Expected Utility: How It Works-Continued

Now I must compare the expected

utility of drilling with the expected
utility of not drilling—that is U(0).
How do I find that?

Principle: To find U(0), compare $0 for certain with a

gamble offering $600 thousand (with probability p) and -
$200 thousand (with probability 1 – p).

The wildcatter finds his preference for $0 by finding the probability p that
leaves him indifferent to the options of $0 and the gamble (drilling).

15-48
 I find my preference for $0 by
finding the probability p that
leaves me indifferent to the
options of $0 and the gamble
(drilling). Say I have determined
it is 0.5.

Then the expected utility of the 50-50 gamble is equal to:

U (0)  (0.5)U (600)  (0.5)U (200)  (0.5)(100)  (0.5)(0)  50

Expected utility rule: The decision maker should

choose the course of action that maximizes his or
her expected utility.

Should the wildcatter drill?

15-49
15-50
 Notes on Preceding Figure
•The utility curve gives us the “certainty equivalent” of a bet with a given
expected value.
•The CE of a bet with an expected value of $200 thousand is $0. That is, the
certainty of obtaining $0 and a bet with an expected value of $200 thousand
have the same expected utility (50).
•A concave utility curve means the decision maker is risk averse—that is,
there is a discount due to risk equal to E(v) – CE ($200 thousand in the
example above).
•A risk-neutral decision maker is guided by expected value—there is no
discount for risk so that the utility curve is linear.
•A risk lover has a convex utility curve.

15-51